An Overview of Various Importance Measures of Reliability System

One of the purposes of system reliability analysis is to identify the weaknesses or the critical components in a system and to quantify the impact of component’s failures. Various importance measures are being introduced by many researchers since 1969. These component importance measures provide a numerical rank to determine which components are more important to system reliability improvement or more critical to system failure. In this paper, we overview various components importance measures and briefly discuss them with examples. We also discuss some other extended importance measures and review the developments in study of various importance measures with respect to some of the popular reliability systems.


Introduction
Reliability evaluation has a vital importance at all stages of processing and controlling in engineering systems. Apart from reliability evaluation of the systems, the study of component importance is also an important part. The component importance (reliability importance index) is valuable in establishing the direction and prioritization of actions related to an upgrading effort (reliability improvement) in system design or suggesting the most effective way to operate and maintain system status. The main purpose of system reliability analysis is to identify the weaknesses/critical components in a system and to quantify the impact of component's failures. Since 1969, various component importance measures are being introduced by the researchers. These measures provide a numerical rank to determine which components are more important (more critical) to system reliability improvement (system failure).
In this paper, we overview various component importance measures for coherent systems. Bergman et al. (1985) provides an interesting survey on reliability theory. Boland and El-Neweihi (1995) overviewed the measures of component importance in binary coherent systems and suggested some new component importance measures which might be useful to analyze when the system is about to undergo a reliability improvement. Kuo and Zhu (2012a, 2012b, 2012c surveyed the recent advances of importance measures in reliability and Zhu and Kuo (2014) gave the implications to mathematical programming, sensitivity and uncertainty analysis, probabilistic risk analysis and probabilistic safety assessment. The review of particular research topic usually helps the researchers in the sense of availability of all related literature in collective form. With this realization, we motivated to present the variety of importance measures altogether. The idea is to bring the importance measures with different bases/ backgrounds in collective form.
These three classes are as; structure importance measures, reliability importance measures and lifetime importance measures. We briefly discuss these three classes of importance measures.

(a) Structure Importance Measures
Structure importance measures are defined under the assumption that the system structure is known. It measures the relative importance of various components with respect to their positions in a system. It is relevant to building a system when several components with distinct reliabilities can be arbitrarily assigned to several locations in the system. Presumably, one would like to assign the most reliable component to the more important location.

(b) Reliability Importance Measures
Reliability importance measures depend on both the system structure and reliability of components over an explicit and fixed mission time. It measures the change in the system reliability with respect to the change in reliability of a specific component.

(c) Lifetime Importance Measures
The lifetime importance measures, depends on both, the position of the component in the system and component lifetime distribution. Kuo and Zhu (2012c) classified the lifetime importance measures into two classes as Time-Dependent Lifetime (TDL) importance and Time Independent Lifetime (TIL) importance, depending on whether they are a function of time.
In this paper, we overview the various importance measures based on the above three classes and also discuss some other extended importance measures. The paper is organized as follows.
In Section 2, we discuss various structure importance measures. Section 3 covers the probability based importance measures and presents examples for some of the popular systems. Section 4 includes a review of lifetime importance measures. In section 5, we overview some other importance measures studied in the literature. Section 6, includes introduction to some extended importance measures. Section 7, reviews the study of importance measures of some weighted systems. Finally, Section 8 concludes the overview of importance measures.

Structure Importance Measures
In this section, we discuss some structure importance measures, which actually evaluate the relative importance of various components with respect to their positions in a system. The component structure importance generally represents the importance of the positions in the system that the components occupy. The structure importance measure is used to assign a rank to the components when the component reliabilities are not available. We discuss some of the structure importance measures in the following. = ( 1 , 2 , … , ): the vector of reliabilities of -components. ( ) the system state which is defined as, ( ) = 1(0) if the system is working (failed).
( ) system reliability with component reliability vector .

(a) Birnbaum Structure (B-Structure) Importance
A structure importance measure is introduced by Birnbaum (1969). The component structure importance generally represents the importance of the positions in the system that the components occupy. Birnbaum (1969) defines the B-structure importance ( )of component as, where (1 , ) : with = 1 (0 , ) : with = 0 (1 , ) : System state when ℎ component is working (0 , ) : System state when ℎ component is fail.
This formula further simplifies as follows.
The B-structural importance establishes the probability of the system failure due to a given component when all components are assumed to be equally reliable i.e. each component has reliability 1 2 .

(b) Fassell-Vesely (FV) Structure Importance
The FV-importance is introduced by Fussell and Vesely (1972) which is based on the cut set and path set. The FV-importance is classified into -type and -type FV-importance. The -type FVimportance, referred to as -FV importance, is based on cut set while the -type FV-importance, referred to as -FV importance, is based on the path set of the system. Let = {1,2, … , } be a set of components of system.
The -FV importance of component is denoted by ( ) and defined as,

(c) Barlow-Proschan (BP) Structure Importance
The BP-structure importance is introduced by . The BP-structure importance of component is denoted by ( ) and defined as, i.e. = 1 2 , = 1,2, … , while the BP-structure importance is average of ( ) over all ∈ [0,1]. Thus, both measures favor neither high nor low component reliabilities. Butler (1979) proposes the two structure importances, namely, cut importance and path importance. The cut importance is a complete ranking of all components relative to their importance to the system and is defined on the basis of minimal cut sets while the path importance is defined on the basis of the minimal path set. The additional coverage of cut and path importances is discussed by Kuo and Zhu (2012c).

(e) Permutation Importance and Permutation Equivalence
The permutation importance and permutation equivalence measures are introduced by Boland et al. (1989). The definition of the permutation importance is based on the minimal cut set and minimal path set.
That is, ( , ) is a vector of − 2 component states excluding ℎ and ℎ component.
If equality holds for some ( , ) , then component and component are said to be permutation equivalent and is denoted by, = .
Component and component are said to be permutation equivalent if and only if they are structurally symmetric and ( ) is said to be a permutation symmetric in and . Koutras et al. (1994) proved the following properties for permutation importance and permutation equivalence.
(i) If > and > then > (transitivity property). (ii) If > for system structure , then > for its dual system structure (dual of ) and vice versa (dual relation).
This implies that component is more permutation important than component i.e. ≥ , for all ≠ (special case property).
In next section, we discuss the various reliability based importance measures.

Reliability Importance Measures
Reliability importance measures depend on both the system structure and reliability of components over an explicit and fixed mission time. It measures the change in the system reliability with respect to the change in reliability of a specific component.
In this section, we discuss the reliability importance measures which are considered when the mission time of a system is implicit and fixed, and consequently, the components are evaluated by their reliability at a fixed time point, i.e. the probability that component functions properly during the mission time (Kuo and Zhu, 2012c).

(a) Birnbaum Reliability Importance
Birnbaum reliability importance (B-importance) measure is introduced by Birnbaum (1969). It is used when we are provided the survival probabilities of all the components in a coherent system. It is the rate of increase in the system reliability with respect to the increase in component reliability. Birnbaum (1969) defines the B-importance of system with component reliabilities = ( 1 , 2 , … , ) through the following.
The B-importance of ℎ component for the functioning of system is denoted by (1 , ) and defined as, The B-importance of ℎ component for the failure of system is denoted by (0 , ) and defined as, Further Birnbaum (1969) defines the B-importance ( )of ℎ component in component system as addition of B-importance for system functioning and system failure. Thus ( ) can be given as, An equivalent definition of B-importance is given by .
is a reliability when ℎ component is working (failed).
Specifically, (3.1) is given under the assumption that component reliabilities are independent.
Birnbaum measure is thus obtained by partial differentiation of the system reliability with respect to when the component reliabilities are independent. Chang et al. (2002) refers this measure as a Combinatorial Birnbaum Importance. According to the equivalent definition (3.1), ( ) is a rate at which system reliability improves, when, the reliability of component improves.
B-importance is the most popular importance measure. We demonstrate the evaluation of Bimportance for series, parallel and -out-of-systems through following examples.
Example 1: A series system works (fails) if and only if all (at least one) components works (fail). The structure function ( ) and reliability function ( ) of a -component series system is given by, The B-importance of ℎ component is, Thus, the most (least) reliable component has the smallest (largest) B-importance.
Example 2: A parallel system, functions (fails) if and only if at least one (all) component functions (fails). The structure and reliability function for this -component system is given by, The B-importance of ℎ component of this system is, That is, the ℎ component with the greatest reliability has the highest B-importance (i.e. most reliable component is most important).

Example 3:
A -out-of-system functions (fails) if and only if at least (1 ≤ ≤ ) of the components function (fail). If = , the -out-of-system reduces to series system and if = 1, it reduces to the parallel system.

 Review of B-Importance of Consecutive-Systems
A linear consecutive-k-out-of-n: F (C(k, n: F)) system is introduced by Kontoleon (1980), and fails if and only if at least k consecutive components in the system fail. Zuo (1993) studies the Bimportance of C(k, n: F) and C(k, n: G) system and claimes that the comparison of B-importance between two components for a C(k, n: G) system is the same as that for the C(k, n: F) system. Hwang et al. (2000) showed that the comparison given by Zuo is not correct and provided a correct relation between the B-importance of components of the two systems. Chang et al. (1999) studied the B-importance of the C(k, n: F) system. Chadjiconstantinidis and Koutras (1999) studied the Bimportance of components for the wide class of Markov Chain Imbeddable Systems (MIS) and provided formulae for the evaluation of the B-importance of the components of an MIS through products of transition probability matrices and multiple recurrence relations. Chang et al. (2002) introduced the concept of uniform, half-line and combinatorial B-importance of components for consecutive-k systems. Meng (1996) compares the importance of system components by the structural characteristics of the system. Kamalja (2012) studies the B-importance of consecutivetype systems as C(k, n: F), m-consecutive-k-out-of-n: F and r-within-consecutive-k-out-of-n: F system. Shen and Cui (2015) studied B-importance for sparsely connected circular consecutive-k systems. Lin et al. (2016) presented the reliability modeling on consecutive-k r -out-of-n r : F linear zigzag structure and circular polygon structured systems. Cai et al. (2016) proposed a Bimportance-based genetic algorithm to search the near global optimal solution for linear consecutive--out-of-system.

(b) Improvement Potential Importance
The B-importance of ℎ component for the system functioning is also known as improvement potential importance of ℎ component (Hoyland and Rausand, 1994;Aven and Jensen, 1999;Freixas and Pons, 2008) or risk achievement importance (van der Borst and Schoonakker, 2001). (1 , ) is the increase in systems' reliability when component is perfect i.e. when = 1 and represents the maximum potential improvement in systems' reliability that can be obtained by improving the reliability of component .
The improvement potential importance ( ) of the ℎ component of the system is the difference between system reliability with perfect component (i.e. component with its reliability = 1) and the system reliability with actual component . That is, (c) Criticality Reliability Importance Kuo and Zuo (2003) proposed the criticality reliability importance measure for system failure (functioning) and is the probability that component fails (functions) and is critical for system failure (functioning) given that the system fails (functions).
The criticality reliability importance ( ) of the ℎ component of system failure given that the system fails is defined as follows.
The criticality reliability importance ( ) of the ℎ component of system ( ) functioning given that the system functions is defined as follows. (d) Bayesian Reliability Importance Birnbaum (1969) proposes the Bayesian reliability importance, which is defined as the probability that component fails given that the system fails. Thus the Bayesian importance ( ) of the ℎ component of system with structure function ( ) is defined as follows. (e) Fussell-Vesely (FV) Importance Fussell and Vesely (1972) introduced a component importance measure using cut set and path set and is referred as FV-cut-importance and FV-path-importance respectively.
The FV-cut-importance (path-importance) of component , denoted by ( ) is defined as the probability that a component state vector has corresponding cut (path) that causes (makes) system failure (function), and contains a minimal cut in ̅ (path in ̅ ) i.e. containing component . Mathematically, We demonstrate it for 2-component series system through following example.
Example 4: Consider a series system with two components with independent reliabilities, 1 = 0.98 and 2 = 0.96. The reliability of the series system is, There is only one cut set containing component 1 and 2, i.e. ̅ = {(1,2)}. Now, FV-cut-importance of component 1 and 2, is as follows.
Example 5: Consider a -component parallel system. For this system constitutes a minimal cut set is ̅ = {1,2, … , }. Hence, FV cut-importance for all components is same. A weakness of this measure is that the same level of importance is calculated for all components in a parallel system regardless of their reliabilities.

Lifetime Importance Measures
The lifetime importance measures depend on both, the position of the component  Lambert (1975) and Natvig (1979) extended the B-importance to the B-lifetime importance at time by using component lifetime distribution.

(a) Birnbaum Lifetime (B-Lifetime) Importance
The B-lifetime importance of component at time is denoted by ( ̅ ( )) and is defined as the probability that the system is in a state at time in which component is critical for the system. That is, the probability that at time the failure and functioning of component coincides with system failure and functioning respectively. Xie (1987) provides the following properties for the system with structure function , lifetime distribution ( ) and reliability distribution ̅ ( ) at time in which a component is in series or parallel with the rest of a system. International Journal of Mathematical, Engineering andManagement Sciences Vol. 2, No. 3, 150-171, 2017 ISSN: 2455-7749 (b) Criticality Lifetime Importance Lambert (1975) defines the criticality lifetime importance for system failure at time , which is relevant to failure diagnosis.The criticality lifetime importance for the system failure of component at time is denoted by ( ̅ ( )) and is defined as, the probability that component has failed by time and component is critical for the system at time , given that the system has failed by time .
Mathematically, Similarly, the criticality lifetime importance for system functioning can be defined. Fussell and Vesely (1972) introduced a component importance measure using minimal cut/path set. Lambert (1975) defines the FV-lifetime cut-importance through a minimal cut set and Kuo and Zhu (2012c) extended the FV-lifetime cut-importance to the FV-lifetime path-importance using a minimal path set.

(c) Fussell-Vesely (FV) Lifetime Importance
The FV-lifetime cut-importance of component at time is denoted by ( ( )) and is defined as the probability that at least one minimal cut containing component fails at time given that the system fails at time . That is, .
Similarly, the FV lifetime path-importance of component at time is denoted by ( ( )) and is defined as, .   ( ( )) can be expressed in terms of B-lifetime importance as follows.

(d) Barlow-Proschan (BP) Lifetime Importance
Thus, ( ( )) is a weighted average of the ( ( )), over an infinite mission time. Boland and El-Neweihi (1995) provided a precise justification that ( ( )) is exactly the probability that the system lifetime coincide with the life of component , that is where is lifetime of system with structure function and is lifetime of component Iyer (1992) extends the Barlow-Proschan index to the more general case when the component lifetimes are jointly absolutely continuous but not necessarily independent. In this setting the index may depend not only on the structure function but also on the distribution function . (e) Natvig Importance Natvig (1979) introduces a new component importance measure known as Natvig's measure. A Natvig's measure has the same characteristics as the FV-lifetime importance.

Properties of BP-Lifetime Importance
If is the reduction in remaining lifetime due to the failure of the ℎ component then a Natvig measure of importance of component is defined as, Xie (1988) extends the relationship of BP-lifetime measure to the Natvig measure as, ( ( )) ≥ ( ( )) for all ≠ , then ≥ for all ≠ .
Aven (1986) compares ( ( )) and by restricting attention to a finite interval of time assuming component lifetimes Weibull distributed. Calculation of is quite feasible for series and parallel systems where proportional hazard modeling for components is appropriate.

Other Importance Measures
In this section, we overview some other importance measures which are apart from the 3-classes defined by Birnbaum (1969) and briefly discuss it.

(a) Hwang Index
Hwang (2001) proposes a new index of component importance based on the cut set, path set, cut and path absoluteness and minimal cut and minimal path set.
Hwang index for component is denoted by ℎ and is defined as, where ( ) is the set of cut sets of cardinality that contains component .

(b) Risk Achievement Worth
Risk Achievement Worth (RAW) has been introduced by Cheok et al. (1998aCheok et al. ( , 1998b for risk analysis terminology. Risk Achievement Worth mainly used as a risk importance measure in probabilistic safety assessments of nuclear power stations. RAW of component at time is denoted by ( ) and defined as the ratio of the system unreliability with component failed at time to the actual system unreliability at time . Mathematically, ( ) = 1− (0 , ( )) 1− ( ( )) . Cheok et al. (1998aCheok et al. ( , 1998b   . This index gives the percentage of times that a restoration of the component will result in restoring the system from a down state in time interval (0, ).

(f) Operational Criticality Index
Operational Criticality Index (OCI) has been introduced by Wang et al. (2004).

Generalized (Pairs and Group of Components) Importance Measures
In this section, we discuss the generalized or pairs/ group of importance measures and also discuss conditional reliability importance introduced in the literature.
The importance measures that are presented in Section 2 to Section 5, evaluate the strength of an individual component. Here, we discuss importance measures for a pair or group of components, considering the effects and higher order interactions of components on system performance. It includes Joint Reliability Importance (JRI) and Joint Failure Importance (JFI) introduced by Hong and Lie (1993). Note that the order of components does not matter for any importance measure of the pair or the group.

(a) JRI and JFI of Two Components
JRI (JFI) of two components is a measure of interaction of two components in a system contribution to the system reliability (failure).
Mathematically, JRI of component 1 , 2 is denoted by ( 1 , 2 ) and for -component system with reliability function ( ) of statistically independent component reliabilities 1 , 2 , … , is defined as, While JFI of 1 , 2 is denoted by ( 1 , 2 ) and for a system with statistically independent component reliabilities is defined as,

and
( 1 , 2 ) = 2 ( ) 1 2 . Armstrong (1995) extends the concept of JRI to include a system with -dependent components and proved that JRI is always non-zero for some classes of systems.
( 1 ) = ∏ =1 , Hong et al. (2002) identified the sign of the JRI of two components without finding its value using Schur-convexity of reliability function and studied JRI for k-out-of-n: G system. Hong et al. (2002) compared the JRI of 2-out-of-n and 2-out-of-(n + 1) system and stated the inequality about JRI of pair of components as 2, ( ) < 2, +1 ( ) when > 1 2 .
The formula for JRI of 2-components of the -out-of-system with i.i.d. component reliabilities for ≥ 3 and 2 ≤ ≤ given by Hong et al. (2002) is as follows. ).

 JRI of Three Components
The JRI of three components 1 , 2 and 3 for -component system with reliability function ( ) and component reliabilities 1 , 2 , … , is given by, For independent component reliabilities, ( 1 , 2 , 3 ) simplifies as follows.
Example 7: The JRI of 3 components for -out-of-: system with i.i.d. component reliabilities and ≥ 3 (given by Gao et al. (2007) is,

 JRI of More Than Three Components
The definition of JRI of 3 components to a group of components is extended as follows.

(b) Conditional Reliability Importance
The conditional reliability importance is introduced by Gao et al. (2007). Under the condition that some components are working or some components are failed, the reliability importance of other components is evaluated. Based on the definition of Marginal Reliability Importance (MRI) and JRI, we consider the reliability importance when the states of certain components are known. It is assumed that the system also works under these conditions. Here we present the definitions of Conditional MRI and Conditional JRI given by Gao et al. (2007).

Review of Importance Measures of Weighted-Systems
In many engineering systems, its components contribute to the system not only through its (working or fail) status, but also through their capacity/load (weight). The components contribute differently to the system performance. Thus, components of a system may be associated with an additional weight characterization apart from its status. In real life, the weight of a component may be its heating/cooling capacity, voltage/wattage, test-score etc. relative to the whole system. This leads to the concept of weighted systems, which consists of components having some positive integer weight.
Wu and Chen (1994a) introduced a weighted -out-of-: ( ) system which consists of components, each one having positive integer weight such that the total system weight is = ∑ =1 and the system works (fail) if and only if the total weight of working (failed) components is at least , a prespecified value. Since is a weight, it may be larger than . Samaniego and Shaked (2008) referred these types of systems as Systems with Weighted Components (SWCs). Further Wu and Chen (1994b) introduced and studied a weighted version of consecutive--out-of-: ( ( , : )) system in linear as well as circular case. A weighted-consecutive--out-of-: system fails if and only if the total weight of failed consecutive components is at least . Chadjiconstantinidis and Koutras (1999) provided formulae for the evaluation of the B-importance and the improvement potential importance of components of weighted k -out-of-n: F and C w (k, n: F) system using Markov chain imbedding technique. Amrutkar and Kamalja (2014) developed formulae for evaluation of reliability and the reliability importance of weighted k-outof-n: Fsystem through Weighted Markov Binomial Distribution (WMBD). Eryilmaz and Bozbulut (2014) studied joint and marginal Birnbaum and Barlow-Proschan importance measures of components of weighted k-out-of-n: G system by using the universal generating function. Rahmani et al. (2016) studied the importance of components in k-out-of-n system with components having random weights. Kamalja and Amrutkar (2014) studied reliability importance of C w (k, n: F) system.

Discussion and Conclusions
In this paper we presented an overview of various importance measures of coherent system. Apart from the importance measures discussed in the present manuscript, there are many other importance measures. This paper is an effort to overview the some of the popular importance measures and study of these for some common systems such as, series, parallel, -out-of-, consecutive and their weighted versions.
In this paper, we overviewed the recent literature on component importance measures. Further, we presented these importance measures for different well-known systems. This survey would spread awareness among readers regarding the importance measures available as well as encourage further research in deciding, which importance measures and when it should be used.